A360284 Least integer nu such that the first zero of the Bessel j-function of index nu is at least nu + n.

0, 2, 7, 16, 29, 48, 73, 106, 148, 199, 260, 333, 417, 515, 627, 754, 897, 1057, 1234, 1431, 1647, 1884, 2142, 2423, 2727, 3056, 3410, 3791, 4198, 4634, 5099, 5594, 6120, 6678, 7268, 7893, 8552, 9247, 9979, 10748, 11555, 12402, 13290

Offset

2,2

Comments

Tricomi proved that the first zero of j_nu occurs at nu + a*nu^(1/3) + b*nu^(-1/3) + O(1/nu). The PARI program below uses an estimate with a = 1.85575708087 and b = 1.

References

Francesco Tricomi, Sulle funzioni di Bellel di ordine e argomento pressochè uguali, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 83:3-20 (1949).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 2..100

Árpád Elbert and Andrea Laforgia, An asymptotic relation for the zeros of Bessel functions, Journal of Mathematical Analysis and Applications, Volume 98, Issue 2 (February 1984), pp. 502-511.

Roger C. McCann, Lower bounds for the zeros of Bessel functions, Proc. Amer. Math. Soc. 64 (1977), pp. 101-103.

Formula

Tricomi (cited in Elbert & Laforgia and McCann) proved that a(n) ~ kn^3. It seems that k is approximately 0.15647199543.

Prog

(PARI) esta(n)=my(a=1.85575708087); ((n+sqrt(n^2-4*a))/2/a)^3
a(n)=if(n==2, return(0)); my(k=esta(n)\1, t=besseljzero(k)-k); if(t<n, while(besseljzero(k++)-k<n, ); k, while(besseljzero(k--)-k>=n, ); k+1)

Crossrefs

Sequence in context: A130883 A375284 A375282 * A005581 A375279 A064468

Adjacent sequences: A360281 A360282 A360283 * A360285 A360286 A360287

Keyword

nonn

Author

Charles R Greathouse IV, Feb 01 2023

Status

approved